Stationary wave solutions to two dimensional viscous shallow water equations: theory of small and large solutions

We study a system of forced viscous shallow water equations with nontrivial bathymetry in two spatial dimensions. We develop a well-posedness theory for small but arbitrary forcing data, as well as for a fixed data profile but large amplitude. In the latter case, solutions may actually fail to exist for large amplitude, but in this case we prove that one of three physically meaningful breakdown scenarios occurs. Through the use of implicit function theorem techniques and a priori estimates, we construct both spatially periodic and solitary (non-periodic but spatially localized) solutions. The solitary case is substantially more complicated, requiring a delicate analysis in weighted Sobolev spaces. To the best of our knowledge, these results constitute the first general construction of stationary wave solutions, large or otherwise, to the viscous shallow water equations and the first general analysis of large solitary wave solutions to any viscous free boundary fluid model.